There was a similar question.
Let $X=(S^2\times S^2)/\mathbb{Z}_4$ where $\mathbb{Z}_4$ acts on $S^2\times S^2$ as $(x,y)\mapsto(-y,x)$.
My question: What are the cohomology rings of $X$ with coefficients $\mathbb{Z}_4$ and $\mathbb{Z}_2$? What is the Bockstein homomorphism $$\beta:H^*(X,\mathbb{Z}_4)\to H^{*+1}(X,\mathbb{Z}_2)?$$
Note that the $\mathbb{Z}_4$ action on $S^2\times S^2$ is orientation-reversing, so $X$ is non-orientable. Also the Euler characteristic is $\chi(X)=1$.
Thank you!